A008864 - OEIS

%I #123 May 11 2024 21:52:52

%S 3,4,6,8,12,14,18,20,24,30,32,38,42,44,48,54,60,62,68,72,74,80,84,90,

%T 98,102,104,108,110,114,128,132,138,140,150,152,158,164,168,174,180,

%U 182,192,194,198,200,212,224,228,230,234,240,242,252,258,264,270,272,278,282,284

%N a(n) = prime(n) + 1.

%C Sum of divisors of prime(n). - _Labos Elemer_, May 24 2001

%C For n > 1, there are a(n) more nonnegative Hurwitz quaternions than nonnegative Lipschitz quaternions, which are counted in A239396 and A239394, respectively. - _T. D. Noe_, Mar 31 2014

%C These are the numbers which are in A239708 or in A187813, but excluding the first 3 terms of A187813, i.e., a number m is a term if and only if m is a term > 2 of A187813, or m is the sum of two distinct powers of 2 such that m - 1 is prime. This means that a number m is a term if and only if m is a term > 2 such that there is no base b with a base-b digital sum of b, or b = 2 is the only base for which the base-b digital sum of m is b. a(6) is the only term such that a(n) = A187813(n); for n < 6, we have a(n) > A187813(n), and for n > 6, we have a(n) < A187813(n). - _Hieronymus Fischer_, Apr 10 2014

%C Does not contain any number of the format 1 + q + ... + q^e, q prime, e >= 2, i.e., no terms of A060800,  A131991, A131992, A131993 etc. [Proof: that requires 1 + p = 1 + q + ... + q^e, or p = q*(1 + ... + q^(e-1)). This is not solvable because the left hand side is prime, the right hand side composite.] - _R. J. Mathar_, Mar 15 2018

%C 1/a(n) is the asymptotic density of numbers whose prime(n)-adic valuation is odd. - _Amiram Eldar_, Jan 23 2021

%D C. W. Trigg, Problem #1210, Series Formation, J. Rec. Math., 15 (1982), 221-222.

%H Reinhard Zumkeller, <a href="/A008864/b008864.txt">Table of n, a(n) for n = 1..10000</a>

%H R. P. Boas and N. J. A. Sloane, <a href="/A005381/a005381.pdf">Correspondence, 1974</a>

%H N. J. A. Sloane and Brady Haran, <a href="https://www.youtube.com/watch?v=6X2D497is6Y">Eureka Sequences</a>, Numberphile video (2021).

%F a(n) = prime(n) + 1 = A000040(n) + 1.

%F a(n) = A000005(A034785(n)) = A000203(A000040(n)). - _Labos Elemer_, May 24 2001

%F a(n) = A084920(n) / A006093(n). - _Reinhard Zumkeller_, Aug 06 2007

%F A239703(a(n)) <= 1. - _Hieronymus Fischer_, Apr 10 2014

%F From _Ilya Gutkovskiy_, Jul 30 2016: (Start)

%F a(n) ~ n*log(n).

%F Product_{n>=1} (1 + 2/(a(n)*(a(n) - 2))) = 5/2. (End)

%p A008864:=n->ithprime(n)+1; seq(A008864(n), n=1..50); # _Wesley Ivan Hurt_, Apr 11 2014

%t Prime[Range[70]]+1 (* _Vladimir Joseph Stephan Orlovsky_, Apr 27 2008 *)

%o (PARI) forprime(p=2,1e3,print1(p+1", ")) \\ _Charles R Greathouse IV_, Jun 16 2011

%o (PARI) A008864(n) = (1+prime(n)); \\ _Antti Karttunen_, Mar 14 2021

%o (Haskell)

%o a008864 = (+ 1) . a000040

%o -- _Reinhard Zumkeller_, Sep 04 2012, Oct 08 2012

%o (Magma) [NthPrime(n)+1: n in [1..70]]; // _Vincenzo Librandi_, Jul 30 2016

%o (Sage) [nth_prime(n) +1 for n in (1..70)] # _G. C. Greubel_, May 20 2019

%Y Cf. A000040, A060800, A131991, A131992, A131993, A141468.

%Y Cf. A007953, A079696, A187813, A239703, A239708.

%Y Column 1 of A341605, column 2 of A286623 and of A328464.

%Y Partial sums of A125266.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, _R. K. Guy_